Step |
Hyp |
Ref |
Expression |
1 |
|
ioorf.1 |
⊢ 𝐹 = ( 𝑥 ∈ ran (,) ↦ if ( 𝑥 = ∅ , 〈 0 , 0 〉 , 〈 inf ( 𝑥 , ℝ* , < ) , sup ( 𝑥 , ℝ* , < ) 〉 ) ) |
2 |
|
ioof |
⊢ (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ |
3 |
|
ffn |
⊢ ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ → (,) Fn ( ℝ* × ℝ* ) ) |
4 |
|
ovelrn |
⊢ ( (,) Fn ( ℝ* × ℝ* ) → ( 𝑥 ∈ ran (,) ↔ ∃ 𝑎 ∈ ℝ* ∃ 𝑏 ∈ ℝ* 𝑥 = ( 𝑎 (,) 𝑏 ) ) ) |
5 |
2 3 4
|
mp2b |
⊢ ( 𝑥 ∈ ran (,) ↔ ∃ 𝑎 ∈ ℝ* ∃ 𝑏 ∈ ℝ* 𝑥 = ( 𝑎 (,) 𝑏 ) ) |
6 |
|
0le0 |
⊢ 0 ≤ 0 |
7 |
|
df-br |
⊢ ( 0 ≤ 0 ↔ 〈 0 , 0 〉 ∈ ≤ ) |
8 |
6 7
|
mpbi |
⊢ 〈 0 , 0 〉 ∈ ≤ |
9 |
|
0xr |
⊢ 0 ∈ ℝ* |
10 |
|
opelxpi |
⊢ ( ( 0 ∈ ℝ* ∧ 0 ∈ ℝ* ) → 〈 0 , 0 〉 ∈ ( ℝ* × ℝ* ) ) |
11 |
9 9 10
|
mp2an |
⊢ 〈 0 , 0 〉 ∈ ( ℝ* × ℝ* ) |
12 |
8 11
|
elini |
⊢ 〈 0 , 0 〉 ∈ ( ≤ ∩ ( ℝ* × ℝ* ) ) |
13 |
12
|
a1i |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ 𝑥 = ∅ ) → 〈 0 , 0 〉 ∈ ( ≤ ∩ ( ℝ* × ℝ* ) ) ) |
14 |
|
simplr |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 𝑥 = ( 𝑎 (,) 𝑏 ) ) |
15 |
14
|
infeq1d |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → inf ( 𝑥 , ℝ* , < ) = inf ( ( 𝑎 (,) 𝑏 ) , ℝ* , < ) ) |
16 |
|
simplll |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 𝑎 ∈ ℝ* ) |
17 |
|
simpllr |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 𝑏 ∈ ℝ* ) |
18 |
|
simpr |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → ¬ 𝑥 = ∅ ) |
19 |
18
|
neqned |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 𝑥 ≠ ∅ ) |
20 |
14 19
|
eqnetrrd |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → ( 𝑎 (,) 𝑏 ) ≠ ∅ ) |
21 |
|
df-ioo |
⊢ (,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } ) |
22 |
|
idd |
⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) → ( 𝑤 < 𝑏 → 𝑤 < 𝑏 ) ) |
23 |
|
xrltle |
⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) → ( 𝑤 < 𝑏 → 𝑤 ≤ 𝑏 ) ) |
24 |
|
idd |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝑎 < 𝑤 → 𝑎 < 𝑤 ) ) |
25 |
|
xrltle |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝑎 < 𝑤 → 𝑎 ≤ 𝑤 ) ) |
26 |
21 22 23 24 25
|
ixxlb |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ∧ ( 𝑎 (,) 𝑏 ) ≠ ∅ ) → inf ( ( 𝑎 (,) 𝑏 ) , ℝ* , < ) = 𝑎 ) |
27 |
16 17 20 26
|
syl3anc |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → inf ( ( 𝑎 (,) 𝑏 ) , ℝ* , < ) = 𝑎 ) |
28 |
15 27
|
eqtrd |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → inf ( 𝑥 , ℝ* , < ) = 𝑎 ) |
29 |
14
|
supeq1d |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → sup ( 𝑥 , ℝ* , < ) = sup ( ( 𝑎 (,) 𝑏 ) , ℝ* , < ) ) |
30 |
21 22 23 24 25
|
ixxub |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ∧ ( 𝑎 (,) 𝑏 ) ≠ ∅ ) → sup ( ( 𝑎 (,) 𝑏 ) , ℝ* , < ) = 𝑏 ) |
31 |
16 17 20 30
|
syl3anc |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → sup ( ( 𝑎 (,) 𝑏 ) , ℝ* , < ) = 𝑏 ) |
32 |
29 31
|
eqtrd |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → sup ( 𝑥 , ℝ* , < ) = 𝑏 ) |
33 |
28 32
|
opeq12d |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 〈 inf ( 𝑥 , ℝ* , < ) , sup ( 𝑥 , ℝ* , < ) 〉 = 〈 𝑎 , 𝑏 〉 ) |
34 |
|
ioon0 |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) → ( ( 𝑎 (,) 𝑏 ) ≠ ∅ ↔ 𝑎 < 𝑏 ) ) |
35 |
34
|
ad2antrr |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → ( ( 𝑎 (,) 𝑏 ) ≠ ∅ ↔ 𝑎 < 𝑏 ) ) |
36 |
20 35
|
mpbid |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 𝑎 < 𝑏 ) |
37 |
|
xrltle |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) → ( 𝑎 < 𝑏 → 𝑎 ≤ 𝑏 ) ) |
38 |
37
|
ad2antrr |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → ( 𝑎 < 𝑏 → 𝑎 ≤ 𝑏 ) ) |
39 |
36 38
|
mpd |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 𝑎 ≤ 𝑏 ) |
40 |
|
df-br |
⊢ ( 𝑎 ≤ 𝑏 ↔ 〈 𝑎 , 𝑏 〉 ∈ ≤ ) |
41 |
39 40
|
sylib |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 〈 𝑎 , 𝑏 〉 ∈ ≤ ) |
42 |
|
opelxpi |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) → 〈 𝑎 , 𝑏 〉 ∈ ( ℝ* × ℝ* ) ) |
43 |
42
|
ad2antrr |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 〈 𝑎 , 𝑏 〉 ∈ ( ℝ* × ℝ* ) ) |
44 |
41 43
|
elind |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 〈 𝑎 , 𝑏 〉 ∈ ( ≤ ∩ ( ℝ* × ℝ* ) ) ) |
45 |
33 44
|
eqeltrd |
⊢ ( ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) ∧ ¬ 𝑥 = ∅ ) → 〈 inf ( 𝑥 , ℝ* , < ) , sup ( 𝑥 , ℝ* , < ) 〉 ∈ ( ≤ ∩ ( ℝ* × ℝ* ) ) ) |
46 |
13 45
|
ifclda |
⊢ ( ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ 𝑥 = ( 𝑎 (,) 𝑏 ) ) → if ( 𝑥 = ∅ , 〈 0 , 0 〉 , 〈 inf ( 𝑥 , ℝ* , < ) , sup ( 𝑥 , ℝ* , < ) 〉 ) ∈ ( ≤ ∩ ( ℝ* × ℝ* ) ) ) |
47 |
46
|
ex |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) → ( 𝑥 = ( 𝑎 (,) 𝑏 ) → if ( 𝑥 = ∅ , 〈 0 , 0 〉 , 〈 inf ( 𝑥 , ℝ* , < ) , sup ( 𝑥 , ℝ* , < ) 〉 ) ∈ ( ≤ ∩ ( ℝ* × ℝ* ) ) ) ) |
48 |
47
|
rexlimivv |
⊢ ( ∃ 𝑎 ∈ ℝ* ∃ 𝑏 ∈ ℝ* 𝑥 = ( 𝑎 (,) 𝑏 ) → if ( 𝑥 = ∅ , 〈 0 , 0 〉 , 〈 inf ( 𝑥 , ℝ* , < ) , sup ( 𝑥 , ℝ* , < ) 〉 ) ∈ ( ≤ ∩ ( ℝ* × ℝ* ) ) ) |
49 |
5 48
|
sylbi |
⊢ ( 𝑥 ∈ ran (,) → if ( 𝑥 = ∅ , 〈 0 , 0 〉 , 〈 inf ( 𝑥 , ℝ* , < ) , sup ( 𝑥 , ℝ* , < ) 〉 ) ∈ ( ≤ ∩ ( ℝ* × ℝ* ) ) ) |
50 |
1 49
|
fmpti |
⊢ 𝐹 : ran (,) ⟶ ( ≤ ∩ ( ℝ* × ℝ* ) ) |